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"Is Paul Newman known for having had problems with alcohol?"

Your query indeed answers "...not widely known..."

"Did Paul Newman have problems with alcoholism?"

"Yes, Paul Newman was open about having struggled with alcoholism"

What's the issue? Perhaps Paul Newman isn't _famous_ ("known") for struggling with alcoholism. But he did struggle with alcoholism.

Your usage of "known for" isn't incorrect, but it's indeed slightly ambiguous.